RESOLUTION OF ALGEBRAICAL EQUATIONS. 23 
al 
highest order in f(p), it is impossible, after Y has once been dis- 
posed of, as above, that it can ever return upon our hands, as it 
might do, if it were a subordinate of any of the principal surds in 
c, ¢,, &c. From the same consideration we selected V, a surd of the 
highest order in ¢, ¢, &c. We may obviously go on in the manner 
described, till we have exhausted all the surds that need to be 
disposed of, in order to make the expression for f (p) altogether an 
integral function of p. 
Def. 6. Let f(p) be an algebraical function of a variable p. In- 
stead of Y,, a surd of the lowest order in f(p), having the index 4, 
write 2,Y, in every place where Y, occurs in f(p) in any of its 
powers, z, being an indefinite m” root of unity. Do in like manner 
with all the other surds of the lowest order. Again, Y, being a surd 
of the order next to the lowest in /( p) thus altered, having + for its 
index, and z, being an indefinite n™ root of unity, write 2,Y, for Y, 
in every place where Y, occurs in the function in any of its powers. 
- Proceed in this way, till modifications of the kind described have 
been made upon all the surds in the function, including those of the 
highest order ; and let the function, after having suffered all these 
changes, become ¢ (p). Denote by ¢,, ¢s, s,...... ; $, 5 the values 
of ¢ (p), not necessarily all unequal, that result from taking all 
the possible values of the indefinite numerical quantities, 2, 22, &¢., 
which have been introduced into the function. These expressions, 
1 $e, &e., may be termed the cognate functions of f (p). 
As it is important that a clear apprehension be formed of the 
manner in which we understand the terms ¢,, ¢,, &c., we subjoin 
illustrative examples. Let 
f(P=C4 Vp) +4 sp)’. 
4 
Then, $ (p)=2.(1+21/p) +2 (1 +2,/P) - 
Here there are, including f ( p), six cognate functions ; 
E 3 
J (pP)=o:=(4+ /P) + +P) ; 
ae $ 
$,=2 (1+ vp) +z (1+) ; 
bye (lt /p) +2 (L+yp) 3 
